Existence theory for steady flows of fluids with pressure and shear rate dependent viscosity, for low values of the power-law index
type: Item
instance of: scholarly article
title: Existence theory for steady flows of fluids with pressure and shear rate dependent viscosity, for low values of the power-law index
zbMATH Open document ID: 1198.35174
DOI: 10.4171/ZAA/1389
author: Miroslav Bulíček
author: V. Fišerová
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2009-10-15T00:00:00Z
review text: The existence of weak solutions of a steady flow of an incompressible, homogeneous, non-Newtonian fluid is established on an open, bounded domain \(\Omega \subset {\mathbb R}^d, d \geq 2\) that is supplemented by no-slip boundary conditions. The Cauchy stress is defined with a pressure and shear-rate dependent viscosity such that non-Newtonian features are incorporated. Under certain assumptions on the constitutively determined part of the Cauchy stress the result is obtained by passing two times to the limit in an \((\epsilon, \eta)\)-approximate system. First the existence of the approximations is sketched. It can be shown with help of a Galerkin approximation; the bounds for the pressure and velocity approximations are calculated. Then the \(\epsilon \rightarrow 0\) limit is presented, where the pressure is decomposed into a strongly convergent and a weakly convergent part to show pointwise convergence of the pressure and the velocity derivatives. Here also Lipschitz approximations of Sobolev functions are considered to obtain these results. Similarly such findings are obtained in the \(\eta \rightarrow 0\) limit, proving the overall statement.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 76D05
zbMATH DE Number: 5615429
zbMATH Keywords: existence theory
zbMATH Keywords: weak solutions
zbMATH Keywords: incompressible steady flow
zbMATH Keywords: pressure and shear dependent material coefficients
zbMATH Keywords: Lipschitz approximation
reviewed by: M. D. Korzec
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/1389
OpenAlex ID: W1995739536
description: scientific article; zbMATH DE number 5615429